Results_olr_precip_brian

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OLR and precip

Brian Map

2. Field1 means and variances: space, time, spacetime

  1. The GRAND (space+time) mean of my x and y: 239.973 W/m2 and 4.7379 (mm/d).
  2. The GRAND (space+time) variance of my x and y: 551.1539 (W/m2)^2 and 12.6278 (mm/d)^2.
    • For OLR, single precision gave me 551.1518 in Matlab or 550.922 in IDL. Better use double!
                • (I made var_n.pro force double precision now)
  3. The GRAND standard deviations are: 23.4766 W/m2, 3.5536 mm/d
  4. The SPATIAL variance of my TIME MEAN longitude section is: 333.94925 (W/m2)^2 and 5.7562 (mm/d)^2
  5. The TEMPORAL variance of my LONGITUDE MEAN time series is: 23.780020 (W/m2)^2 and 0.94054618 (mm/d)^2

3. (the assignment part)

  1. Confirm that the time mean of the anomalies as defined above is 0. Yes
  2. Is the spatial mean of the anomalies (as defined above) 0?
      • No
    • Is it the same as the time series of the spatial mean of the raw data? Or is it a new thing?
      • Its graph (a time series) looks kind of like the spatial mean of x, but it is not quite the same. For one thing the grand mean of anomx is 0 while the grand mean of x is not. But even accounting for this simple offset, a time series plot of the difference is a piecewise constant curve that varies from one year to the next. Some students will upload and show plots that make this clear, and hopefully explain it better with math in some notation (maybe handwritten on the board) that helps us see it!
  3. My CLIMATOLOGICAL ANNUAL CYCLE have variance:
    • var( climx12(:),1) 379.7256 (OLR)
      var( climy12(:),1) 7.0662 (precip)
  4. My INTERANNUAL ANOMALY ARRAYS have variance:
    • var( anomx(:),1) 171.4277
      var( anomy(:),1) 5.5616
  5. Fill out a variance decomposition table for field 1: feel free to add columns if you can define other parts.

 field1 name
a) total variance of x
 551.1539
b) purely spatial (variance of TIME mean at each lon)
 333.94925
c) variance of (x minus its TIME mean at each lon)
d) purely temporal (variance of LON mean at each time)
 23.780020
e) variance of (x minus its LON mean at each time)
f) remove both means (space-time variability)
g) mean seasonal cycle
 410.563
h) deseasonalized anomalies
 140.590
i) variance of longitudinal mean of h
 6.4778
j) h minus i
 134.113
6. Discuss your results:

4. Further decomposition of anomx by scale (using rebinning).

    • What space and time scales (units: degrees and months) have the most variance in your anomx field?
      • olr_rebintimelon_variancedist.gif shows that the total variance decreases faster with time rebinning (averaging) than space rebinning, indicating more small-scale structure in time than space.
      • Spatial averaging (2,4,8 rebin factors) doesn't destroy variance as quickly because the structure in space is pretty large scale. But then, suddenly, rebinning by 16 or 48 in longitude suddenly reduces variance a lot, because there are compensating +/- anomalies across longitudes.
      • The above agrees with the structure seen in the raw data for olr anomalies: it is pretty large scale in space. [[file/view/olr_timeanomalies_lonanomalies.gif|olr_timeanomalies_lonanomalies.gif]]
      • Matlab prints it:
        • variance_by_scalefactor =
          140.5907 138.5684 134.0453 124.3102 103.1921 29.0789
          111.8484 110.6262 107.7468 101.2612 86.3631 22.2076
          90.3986 89.6423 87.7923 83.4635 72.0277 16.7211
          68.5817 68.0969 66.8513 63.8972 55.5564 12.5901
          37.5201 37.2371 36.5120 34.8221 29.8212 7.6441
          11.9013 11.8029 11.5493 10.8537 9.1045 2.7723
    • Create a contour plot of anomx -- can you see these "characteristic" scales by eye? Annotate your anomx plot with some ovals of about the right size (in powerpoint may be easiest), and put a few of these ovals in regions where you think you can see structures of about the right scales. I am looking for good eyeball judgement here.
      • Here's a figure illustrating why the variance drops most steeply from 16 to 48 rebinning factor in time: basically, 48 month averaging kills ENSO variance, while 16 month averaging doesn't. olr_rebintimelon_1648check.gif
      • Here's a figure illustrating why the variance drops most steeply from 16 to 48 rebinning factor in time: basically, 48 month averaging kills ENSO variance, while 16 month averaging doesn't. olr_rebintimelon_1648check2.gif


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6. Covariance matrix stuff


HW3_olr_precip_mapes.m has all my matlab code, including the stuff that makes these Matlab pictures:
OLR_anoms_covar_correl.BEM.Matlab.png
OLR.lagregression.BEM.jpg





UNDER CONSTRUCTION - OLD STUFF - IGNORE FOR NOW

5. Autocorrelation and integral scales

  1. Plot the temporal autocorrelation of x at a longitude in the east Pacific. (this is a line plot as a function of time lag).
    • What is the integral time scale (area under the curve above)?
      • It is rather hard to estimate because of all the annual wiggles in the autocorrelation curve
      • Removing the seasonal cycle helps especially over land
  2. Plot the spatial autocorrelation of your x field at a couple of times (or the average of all times).
  3. Take the time series at grid point 85 (the central Pacific) and lag-correlate it with the time series at all other longitudes. Plot the result as a contoured lag vs. longitude diagram. What does this show you that space and time autocorrelations alone didn't?

6. Cross-correlation between your 2 variables

  1. Scatter plot your x and y arrays (all values). Is there a relationship?
  2. What is the correlation coefficient implied by this all-space-and-time=points scatter plot? rho = cov(x(:),y(:)) / sqrt( cov(x(:)) * cov(x(:)) ) rho = correlate(x,y)
    • How much of the variance of y can be 'explained' by linear regression on x (y = mx + b)? How does this relate to rho? What is m?
    • How much of the variance of x can be 'explained' by linear regression on variable y? (x = nx + a)? How does this relate to rho? What is n?
  3. Plot the temporal correlation between your x and variables as a function of longitude. Interpret the results.
  4. Plot the longitudinal correlation as a function of time. Interpret the results.
  5. Plot the cross correlation between x and y for the east Pacific and South American longitudes from problem 5.1 above. This is the correlation between the 2 variables as a function of time lag. Interpret the results. Matlab: xcov() or xcorr()IDL: C-correlate()

Extra credit:
  • Use 1-month lagged correlations and cross correlations like the above to develop a 2-term formula like
    • y_fcst_EPAC(t) = c1 x(t-1) + c2 y(t-1)
  • which gives the best 1-month forecasts of eastern Pacific y you can make, given your x and y data. (multiple regression)